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Relative $$p$$-adic Hodge theory: foundations. (English) Zbl 1370.14025
Astérisque 371. Paris: Société Mathématique de France (SMF) (ISBN 978-2-8569-807-7/pbk). 239 p. (2015).
The aim of this paper is to give a new approach to the relative analogue of the theory of $$(\varphi,\Gamma)$$-modules. In the classical setting, $$(\varphi,\Gamma)$$-modules are modules over certain rings of power series, equipped with a semi-linear Frobenius map $$\varphi$$ and a compatible action of a group $$\Gamma$$. They were defined by J.-M. Fontaine [Prog. Math. 87, 249–309 (1990; Zbl 0743.11066)] who used them to describe the $$p$$-adic representations of the Galois group of a $$p$$-adic field. A key input in Fontaine’s work is the theory of the field of norms, which gives an isomorphism between a subgroup of the Galois group of a $$p$$-adic field and the Galois group of a local field of characteristic $$p$$.
In the relative setting, the Galois group of a $$p$$-adic field is replaced by the étale fundamental group $$G$$ of an affinoid space over a finite extension of $$\mathbb{Q}_p$$. One is still interested in the representations of $$G$$ over finite-dimensional $$\mathbb{Q}_p$$-vector spaces. Thanks to the work of many people (including Andreatta, Brinon, Faltings and Scholl), it progressively became clear that these representations of $$G$$ could be studied using a suitable generalization of Fontaine’s $$(\varphi,\Gamma)$$-modules. This paper provides such a generalization, relying heavily on the theory of Witt vectors and the geometric spaces attached to them.
In addition to the introduction, the paper has nine chapters. The introduction is informative and provides a good overview of the contents of the paper as well as a discussion of the relations between the theory developed in this paper, Scholze’s theory of perfectoïd spaces, and Fargues-Fontaine’s curve in $$p$$-adic Hodge theory. The first chapter contains some algebro-geometric preliminaries. The second chapter discusses the spectra of nonarchimedean Banach rings. In the paper, both the Gelfan’d spectrum and the adic spectrum are used. The third chapter is about strict $$p$$-rings, and contains the generalization of the theory of the field of norms, namely the perfectoïd correspondence. The fourth chapter is about Robba rings and slope theory, and the fifth chapter is about relative Robba rings. The sixth chapter concerns $$\varphi$$-modules and includes a discussion of the relation with Fargues and Fontaine’s constructions. The seventh chapter is about slope filtrations in families, and the eighth is about perfectoïd spaces. The ninth chapter contains the construction of the relative analogue of the theory of $$(\varphi,\Gamma)$$-modules and the various equivalences of categories that one gets. The classical $$(\varphi,\Gamma)$$-modules are replaced by sheaves of $$\varphi$$-modules over rings of period sheaves. The sheaf axiom (for the pro-étale topology) replaces the classical action of $$\Gamma$$.
The last paragraph of the introduction contains a list of “further goals” that the authors plan to work on.

##### MSC:
 14G22 Rigid analytic geometry 11S15 Ramification and extension theory 13J07 Analytical algebras and rings 13F35 Witt vectors and related rings 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14F30 $$p$$-adic cohomology, crystalline cohomology
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